A New Adaptive Square-Root Unscented Kalman Filter for Nonlinear Systems with Additive Noise

Yong Zhou,Yufeng Zhang,Chao Zhang,Juzhong Zhang

doi:10.1155/2015/381478

Abstract

The Kalman filter (KF), extended KF, and unscented KF all lack a self-adaptive capacity to deal with system noise. This paper describes a new adaptive filtering approach for nonlinear systems with additive noise. Based on the square-root unscented KF (SRUKF), traditional Maybeck’s estimator is modified and extended to nonlinear systems. The square root of the process noise covariance matrixQor that of the measurement noise covariance matrixRis estimated straightforwardly. Because positive semidefiniteness ofQorRis guaranteed, several shortcomings of traditional Maybeck’s algorithm are overcome. Thus, the stability and accuracy of the filter are greatly improved. In addition, based on three different nonlinear systems, a new adaptive filtering technique is described in detail. Specifically, simulation results are presented, where the new filter was applied to a highly nonlinear model (i.e., the univariate nonstationary growth model (UNGM)). The UNGM is compared with the standard SRUKF to demonstrate its superior filtering performance. The adaptive SRUKF (ASRUKF) algorithm can complete direct recursion and calculate the square roots of the variance matrixes of the system state and noise, which ensures the symmetry and nonnegative definiteness of the matrixes and greatly improves the accuracy, stability, and self-adaptability of the filter.

Highlights

In 1960, Kalman described a recursive minimum meansquare estimation (RMMSE) solution of the linear discretetime dynamical system filter problem [1]
The unscented Kalman filter (UKF) is more accurate, more stable, and far easier to implement than the extended Kalman filter (EKF) [6,7,8], as proven in aircraft state estimation [9], unmanned aerial vehicle (UAV) attitude estimation [10], rotorcraft UAV actuator failure estimation [11], and satellite attitude estimation [12], among other fields
The UKF was extended to the squareroot unscented Kalman filter (SRUKF) by van der Merwe and Wan [13]

Summary

Introduction

In 1960, Kalman described a recursive minimum meansquare estimation (RMMSE) solution of the linear discretetime dynamical system filter problem [1]. The EKF applies the standard linear Kalman filter methodology by linearizing the true nonlinear system This approach is suboptimal and can lead to divergence. There are few applications for Maybeck’s algorithm in the nonlinear field To overcome these shortcomings, we developed the concept of a new adaptive square-root unscented Kalman filter (ASRUKF) in 2013 [22]. In this study, we detailed the ASRUKF for nonlinear systems with additive noise by extending the SRUKF with the traditional Maybeck estimator to nonlinear systems and tested its effectiveness. To achieve these objectives, a new ASRUKF technique is described in detail based on three different nonlinear systems

The AKF Based on the Maybeck’s Estimator

Simulation and Verification

Conclusions